where ψ is a function of x, y, called the stream- or current-function; interpreted physically, ψ − ψ0, the difference of the value of ψ at a fixed point A and a variable point P is the flow, in ft.3/second, across any curved line AP from A to P, this being the same for all lines in accordance with the continuity.
Thus if dψ is the increase of ψ due to a displacement from P to P′, and k is the component of velocity normal to PP′, the flow across PP′ is dψ = k·PP′; and taking PP′ parallel to Ox, dψ = vdx; and similarly dψ= −udy with PP′ parallel to Oy; and generally dψ/ds is the velocity across ds, in a direction turned through a right angle forward, against the clock.
In the equations of uniplanar motion
| 2ζ = | dv | − | du | = | d2ψ | + | d2ψ | = −∇2ψ, suppose, |
| dx | dy | dx2 | dy2 |
so that in steady motion
| dH | + ∇2ψ | dψ | = 0, | dH | + ∇2ψ | dψ | = 0, | dH | + ∇2ψ = 0, |
| dx | dx | dy | dy | dψ |
and ∇2ψ must be a function of ψ.
If the motion is irrotational,
| u = − | dφ | = − | dψ | , v = − | dφ | = | dψ | , |
| dx | dy | dy | dx |
so that ψ and φ are conjugate functions of x and y,
or putting
The curves φ = constant and ψ = constant form an orthogonal system; and the interchange of φ and ψ will give a new state of uniplanar motion, in which the velocity at every point is turned through a right angle without alteration of magnitude.
For instance, in a uniplanar flow, radially inward towards O, the flow across any circle of radius r being the same and denoted by 2πm , the velocity must be m /r , and
m log reiθ, w = m log z.
Interchanging these values
gives a state of vortex motion, circulating round Oz, called a straight or columnar vortex.
A single vortex will remain at rest, and cause a velocity at any point inversely as the distance from the axis and perpendicular to its direction; analogous to the magnetic field of a straight electric current.
If other vortices are present, any one may be supposed to move with the velocity due to the others, the resultant stream-function being
the path of a vortex is obtained by equating the value of ψ at the vortex to a constant, omitting the r m of the vortex itself.
When the liquid is bounded by a cylindrical surface, the motion of a vortex inside may be determined as due to a series of vortex-images, so arranged as to make the flow zero across the boundary.
For a plane boundary the image is the optical reflection of the vortex. For example, a pair of equal opposite vortices, moving on a line parallel to a plane boundary, will have a corresponding pair of images, forming a rectangle of vortices, and the path of a vortex will be the Cotes’ spiral
this is therefore the path of a single vortex in a right-angled corner; and generally, if the angle of the corner is π/n, the path is the Cotes’ spiral
A single vortex in a circular cylinder of radius a at a distance c from the centre will move with the velocity due to an equal opposite image at a distance a2/c, and so describe a circle with velocity
Conjugate functions can be employed also for the motion of liquid in a thin sheet between two concentric spherical surfaces; the components of velocity along the meridian and parallel in colatitude θ and longitude λ can be written
| dφ | = | 1 | dψ | , | 1 | dψ | = − | dψ | , | ||
| dθ | sin θ | dλ | sin θ | dλ | dθ |
and then
28. Uniplanar Motion of a Liquid due to the Passage of a Cylinder through it.—A stream-function ψ must be determined to satisfy the conditions
so that, if the solid is moving with velocity U in the direction Ox, dψ/ds = −U dy/ds, or ψ + Uy = constant over the moving cylinder; and ψ + Uy = ψ′ is the stream function of the relative motion of the liquid past the cylinder, and similarly ψ − Vx for the component velocity V along Oy; and generally
is the relative stream-function, constant over a solid boundary moving with components U and V of velocity.
If the liquid is stirred up by the rotation R of a cylindrical body,
| = −Rx | dx | − Ry | dy | , |
| ds | ds |
a constant over the boundary; and ψ′ is the current-function of the relative motion past the cylinder, but now
throughout the liquid.
Inside an equilateral triangle, for instance, of height h,
where α, β, γ are the perpendiculars on the sides of the triangle.
In the general case ψ′ = ψ + Uy − Vx + 12R (x2 + y2) is the relative stream function for velocity components, U, V, R.
29. Example 1.—Liquid motion past a circular cylinder.
Consider the motion given by
so that
| ψ = U ( r + | a2 | ) cos θ = U ( 1 + | a2 | ) x, |
| r | r 2 |
| φ = U ( r + | a2 | ) sin θ = U ( 1 + | a2 | ) y. |
| r | r 2 |
Then ψ = 0 over the cylinder r = a, which may be considered a fixed post; and a stream line past it along which ψ = Uc, a constant, is the curve
| ( r − | a2 | ) sin θ = c, (x2 + y2) (y − c) − a2y = 0 |
| r |
a cubic curve (C3).
Over a concentric cylinder, external or internal, of radius r = b,
| ψ′ = ψ + U1y = [ U ( 1 − | a2 | ) + U1] y, |
| b2 |
and ψ′ is zero if
so that the cylinder may swim for an instant in the liquid without distortion, with this velocity U1, and ω in (1) will give the liquid motion in the interspace between the fixed cylinder r = a and the concentric cylinder r = b, moving with velocity U1.
When b = 0, U1 = ∞; and when b = ∞, U1 = −U, so that at infinity the liquid is streaming in the direction xO with velocity U.
If the liquid is reduced to rest at infinity by the superposition of an opposite stream given by ω = −Uz, we are left with
giving the motion due to the passage of the cylinder r = a with velocity U through the origin O in the direction Ox.
If the direction of motion makes an angle θ′ with Ox,
| tan θ′ = | dφ | / | dφ | = | 2xy | = tan 2θ, θ = 12θ′, |
| dy | dx | x2 − y2 |
and the velocity is Ua2/r 2.
Along the path of a particle, defined by the C3 of (3),
| sin2 12θ′ = | y2 | = | y (y − c) | , |
| x2 + y2 | a2 |
| 12 sin θ′ | dθ′ | = | 2y − c | dy | , | |
| ds | a2 | ds |
on the radius of curvature is 14a2/(y − 12c), which shows that the curve
is an Elastica or Lintearia. (J. C. Maxwell, Collected Works, ii. 208.)
If φ1 denotes the velocity function of the liquid filling the cylinder r = b, and moving bodily with it with velocity U1,
and over the separating surface r = b
| φ | = − | U | ( 1 + | a2 | ) = | a2 + b2 | , |
| φ1 | U1 | b2 | a2 ~ b2 |
and this, by § 36, is also the ratio of the kinetic energy in the annular
interspace between the two cylinders to the kinetic energy of the
liquid moving bodily inside r = b.
Consequently the inertia to overcome in moving the cylinder r = b, solid or liquid, is its own inertia, increased by the inertia of liquid (a2 + b2)/(a2 ~ b2) times the volume of the cylinder r = b; this total inertia is called the effective inertia of the cylinder r = b, at the instant the two cylinders are concentric.
